How can I become a piercing apprentice?

I would call your local tattoo shops and ask them for advice. Tattooing and piercing generally go hand-in-hand these days.

My daughter apprenticed at a local tattoo parlor. Normally, you have to pay for such training, but she didn’t, because she is so personable.
The cost is a fee, plus every piercing you do for the first time is free. It will be a while before you make any money doing it.